Compact all-optical precision-tunable narrowband hard Compton X-ray source

Readily available bright X-ray beams with narrow bandwidth and tunable energy promise to unlock novel developments in a wide range of applications. Among emerging alternatives to large-scale and costly present-day radiation sources which severely restrict the availability of such beams, compact laser-plasma-accelerator-driven inverse Compton scattering sources show great potential. However, these sources are currently limited to tens of percent bandwidths, unacceptably large for many applications. Here, we show conceptually that using active plasma lenses to tailor the electron bunch-photon interaction, tunable X-ray and gamma beams with percent-level bandwidths can be produced. The central X-ray energy is tunable by varying the focusing strength of the lens, without changing electron bunch properties, allowing for precision-tuning the X-ray beam energy. This method is a key development towards laser-plasma-accelerator-driven narrowband, precision tunable femtosecond photon sources, enabling a paradigm shift and proliferation of compact X-ray applications.


A. Tunability through APL current
Changing the plasma lens current I APL varies the focusing strength k ∝ I APL of the APL. Thus, at a given interaction plane with the Inverse Compton Scattering (ICS) laser, the focused electron energy slice can be tuned via I APL .
Calculation of the full beam transport, i.e. the Courant-Snyder Parameters (CSPs), from the accelerator output to the ICS interaction relies on the free drift matrix M FD and the beam transport matrix through the active plasma lens M APL given by is the focussing strength of the APL. With z u the distance from the electron source to the APL and α, γ the CSPs at the APL exit, the distance to focus after the APL z v is given by for sufficiently small φ. Assuming linear focussing fields with a field gradient ∂B φ /∂r = µ 0 I APL /(2πr 2 AP L ), with r AP L being the radius of the plasma lens, Eq.(4) can be solved for γ e , yielding the focused electron energy at the distance z v from the APL exit: where z f = z u + L + z v is the interaction plane and K = qµ 0 /(2πm e c). The electron energy focused at a fixed interaction plane thus depends linearly on the APL current I APL and the X-ray energy E x ∝ I 2 APL . The tuning range can be adjusted via the APL length L, radius r AP L , position z u and the interaction plane z f . Fig. 2 shows the good agreement of Eq. (6) with the matrix formalism for lenses of r AP L = 1 mm and r AP L = 2 mm.

B. Derivation of the effective energy spread
In ICS, photon emission depends on the overlap of electrons and laser pulse [2,3]. The smaller the mean bunch size compared to the laser waist, the larger the mean laser field acting on the electrons, and thus the emitted photon number. Due to chromatic focusing of the APL, each energy slice (γ e ) is focused at a different position f (γ e ). Bunch slices of energies deviating from the target electron energy are focused before or after the laser focus. Consequently, they have a larger size in the interaction center and emit less photons. The effective energy spread is defined as the width of N x (γ e ), the number of emitted photons by an electron of energy γ e .
The number of photons emitted in an ICS interaction of a laser with pulse duration τ L and a bunch with charge Q is with the mean effective laser strength parameter a 0,eff . For sufficiently small Thomson interaction lengths compared to the electron bunch beta function the bunch size stays approximately constant during the interaction.
for negligible bunch length cτ b cτ L . This is a valid approximation, assuming typical parameters as in the manuscript [1]. Consequently, longitudinal effects can be neglected and the effective laser strength and photon number can be determined via the two-dimensional overlap on the laser focus. For Gaussian transverse laserintensity and electron-bunch profiles, and I ∝ a 2 : Weighting the intensity distribution with the spatial electron distribution gives the effective laser strength parameter With the beta function β 0 in the slice focus f (γ e ) and the drift length ∆z between the focal point for a given electron energy γ e and the target-energy focus (f (γ e,t ) = f L ), the bunch size is: For sufficiently large γ e and small ∆γ e , the square of the electron slice size is approximately symmetric and of the form σ 2 e (γ e ) ∝ γ 2 e + x, with variable x. Thus, N eff (γ e ) can be described by a Lorentzian and the FWHM width is the effective electron energy spread.

C. Tunability and bandwidth of APL-tunable X-ray source
A shown above, the current variation in the APL accommodates energy tuning of the X-ray beam. The current dependence of central X-ray energy and bandwidth is plotted in Fig. 3a for r AP L = 1 mm, z 0 = 5 cm and z f = 1 m for a beam with energy-independent normalised emittance 0 = 1 µm. The scattered X-ray energy varies from 0.05 MeV to 4.8 MeV, a tuning range of nearly 2 orders of magnitude. The relative X-ray bandwidth emitted into a 1 mrad cone increases by a factor of 6 over this tuning range. The black line in Fig. 3a is the inherent ICS bandwidth κ = γ 2 θ 2 c /(1 + γ 2 θ 2 c ) due to the θ c = 1 mrad collimation angle, highlighting that the bandwidth is limited by the observation angle and other contributions to the bandwidth have been effectively reduced.
For a given plasma lens current range, the X-ray tuning range can be shifted to lower energies by increasing the APL radius and thus reducing the APL strength k ∝ r −2 AP L . As an example, in Fig. 3b the X-ray beam properties are plotted for a plasma lens with radius r AP L = 2 mm. Without any other changes, the tuning range is now from 3 keV to 300 keV, with the bandwidth increasing by a factor of 2.5, mainly due to a higher divergence. As the synchrotron angle 1/γ e 1 mrad, the bandwidth is dominated by contributions from the electron bunch properties and not the observation cone.
The particle tracking code astra [4] was used to calculate the electron trajectories taking into account space charge forces. An electron bunch, represented by 5 × 10 3 macroparticles, is propagated through the linear radial magnetic field of the plasma lens, and, in the interaction plane, through the time-varying electromagnetic fields of the bi-Gaussian (time and space) scattering laser. The classical radiation code Clara2 [5] is employed to determine the photon angular spectrum. Clara2 employs Lienard-Wiechert potentials to calculate the radiation emission on the basis of the macroparticle trajectories.

D. Oscillations in the capillary -multiple energies focused
For sufficient APL length L and focusing strength, low-energy electrons can be "focused" within the capillary, i.e. the spot size of that energy slice "oscillates". As a consequence, electrons of different γ e focus to the same plane (see also Fig 2). This is illustrated in Fig. 4. This effect is negligible on the X-ray spectrum due to various reasons.
Firstly, low γ e electrons obtain a larger divergence at given focusing strength k of the lens, and emit radiation into a larger synchrotron angle θ s ∼ γ −1 e than high γ e electrons, reducing their photon emission into the observation cone. Secondly, the longitudinal separation ∆f /∆γ e is larger, reducing ∆Q/∆z, i.e. the effective number of electrons interacting with the laser. Thirdly, as shown in the example in Fig. 4, the X-ray energies are orders of magnitude apart (E X (γ = 44) ∼ 12 keV, E X (γ = 443) ∼ 1.2 MeV), easily separated via appropriate filtering. For cases such as illustrated in Fig 4, a large fraction of electrons of low γ e will not be focussed to z f by the APL as they will leave the focussing fields within the APL when their excursion exceeds the lens radius. Finally, the low γ e contribution to the X-ray spectrum will be determined and is thus controllable by the number of electrons in that spectral range.